| Exam Board | OCR MEI |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2016 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared goodness of fit |
| Type | Chi-squared goodness of fit: Given ratios |
| Difficulty | Standard +0.3 This is a straightforward application of two standard statistical tests (chi-squared goodness of fit and Wilcoxon signed rank) with clear instructions and given ratios/hypotheses. Both parts require routine application of learned procedures with minimal problem-solving or conceptual insight, making it slightly easier than average for Further Maths Statistics. |
| Spec | 5.06a Chi-squared: contingency tables5.07b Sign test: and Wilcoxon signed-rank |
| Phenotype | Brown body Red eye | Brown body Brown eye | Black body Red eye | Black body Brown eye |
| Frequency | 125 | 37 | 32 | 6 |
| Relative proportion from model | 9 | 3 | 3 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0\): The (genetic) model fits the data. \(H_1\): The (genetic) model does not fit the data | B1 | Both hypotheses; Not 'data fits model' |
| Observed: 125, 37, 32, 6; Expected: 112.5, 37.5, 37.5, 12.5 | B1 | Expected values correct |
| Contributions: 1.3889, 0.0067, 0.8067, 3.38 | M1, A1 | use of \(\frac{(O-E)^2}{E}\) (at least one correct); all correct to 3dp where appropriate |
| \(X^2 = 5.582\) | A1 | cao (3sf or 4sf) |
| Degrees of freedom \(= 3\) | B1 | no FT if wrong (can be implied by 9.348) |
| Critical value \(= 9.348\) | B1 | no FT if wrong |
| \(5.582 < 9.348 \rightarrow\) cannot reject \(H_0\) | M1 | FT their \(X^2\) |
| The data give no reason to doubt the genetic model | E1 | Do not accept 'data fits model'; but 'Evidence suggests that model fits the data' is fine |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: m = 2.5\); \(H_1: m > 2.5\) | B1 | both hypotheses |
| where \(m\) is the population median length (of South American fruit flies) | B1 | definition including median, population, and context |
| Subtract 2.5, ranking with all ranks correct (table shown) | M1, M1, A1 | |
| \(W_- = 16, W_+ = 62\) | B1 | for either, cao |
| \((n=12)\), Critical value \(= 17\); \((16 < 17 \rightarrow)\) reject \(H_0\) | B1 | allow 61 if compared to 62. No FT if wrong |
| Suggests population median length of South American fruit flies exceeds 2.5cm | M1, A1 | FT their \(W\); including median (or 'on average') and context |
# Question 2:
## Part a:
$H_0$: The (genetic) model fits the data. $H_1$: The (genetic) model does not fit the data | B1 | Both hypotheses; Not 'data fits model'
Observed: 125, 37, 32, 6; Expected: 112.5, 37.5, 37.5, 12.5 | B1 | Expected values correct
Contributions: 1.3889, 0.0067, 0.8067, 3.38 | M1, A1 | use of $\frac{(O-E)^2}{E}$ (at least one correct); all correct to 3dp where appropriate
$X^2 = 5.582$ | A1 | cao (3sf or 4sf)
Degrees of freedom $= 3$ | B1 | no FT if wrong (can be implied by 9.348)
Critical value $= 9.348$ | B1 | no FT if wrong
$5.582 < 9.348 \rightarrow$ cannot reject $H_0$ | M1 | FT their $X^2$
The data give no reason to doubt the genetic model | E1 | Do not accept 'data fits model'; but 'Evidence suggests that model fits the data' is fine
## Part b:
$H_0: m = 2.5$; $H_1: m > 2.5$ | B1 | both hypotheses
where $m$ is the population median length (of South American fruit flies) | B1 | definition including median, population, and context
Subtract 2.5, ranking with all ranks correct (table shown) | M1, M1, A1 |
$W_- = 16, W_+ = 62$ | B1 | for either, cao
$(n=12)$, Critical value $= 17$; $(16 < 17 \rightarrow)$ reject $H_0$ | B1 | allow 61 if compared to 62. No FT if wrong
Suggests population median length of South American fruit flies exceeds 2.5cm | M1, A1 | FT their $W$; including median (or 'on average') and context
---2
\begin{enumerate}[label=(\alph*)]
\item A genetic model involving body colour and eye colour of fruit flies predicts that offspring will consist of four phenotypes in the ratio $9 : 3 : 3 : 1$.
A random sample of 200 such offspring is taken. Their phenotypes are found to be as follows.
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
Phenotype & Brown body Red eye & Brown body Brown eye & Black body Red eye & Black body Brown eye \\
\hline
Frequency & 125 & 37 & 32 & 6 \\
\hline
Relative proportion from model & 9 & 3 & 3 & 1 \\
\hline
\end{tabular}
\end{center}
Carry out a test, using a $2.5 \%$ level of significance, of the goodness of fit of the genetic model to these data.
\item The median length of European fruit flies is 2.5 mm . South American fruit flies are believed to be larger than European fruit flies. A random sample of 12 South American fruit flies is taken. The flies are found to have the following lengths (in mm).\\
$1.7 \quad 1.4$\\
$3.1 \quad 3.5$\\
3.8\\
4.2\\
2.2\\
2.9\\
4.4\\
2.6\\
$3.9 \quad 3.2$
Carry out a Wilcoxon signed rank test, using a $5 \%$ level of significance, to test this belief.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI S3 2016 Q2 [18]}}